00:04
So we are given a polynomial x cubed plus 6x minus 20, right? this is our f of x.
00:14
And we are also told that 2 is a 0 of the equation, right? so 2 is a 0.
00:23
So this means that the factor x minus 2 is a factor of this equation, right? so if this is a linear factor, it means that, we need to multiply this linear factor with the quadratic polynomial so that we get this cubic polynomial, right? so we're going to say x minus 2 times in a non -quadratic polynomial that is a x squared plus bx plus c, right? this is going to be equal to x cubed plus 6x minus 20 right so a b and c are unknowns right so now all we need to do is to expand our left hand side and then we equate the coefficients of the x cubed term to one equate the coefficients of the x squared term to six sorry of the x term not the x squared because the coefficients of the x squared is zero in this case and then we equate the constant to minus 20 right so to expand this we're going to get a x cubed right x times a x squared that's a x cubed then x times bx that's plus bx squared then x squared then x times plus c that's plus cx and then minus two times a x squared you get minus two a x squared minus two x squared minus two times plus bx is minus two bx minus two times plus c is minus two c so we can just say this is equal to f of x right and then now we collect like terms so you here we have a x cubed that's just the one term there's no other one with x cubed and then for x squared we're going to have we're going to put the coefficients in brackets now so we have bx squared we put the b here and then we look for any other term that has x squared in it and we find here there is minus 2a right so it's going to be b minus 2a right x squared and and then we do the same thing for the x term.
03:28
We're going to have c minus 2b times x.
03:35
And then our constant is just minus 2c, right? so this is equal to x cubed plus 6x minus 20.
03:51
Right.
03:52
Then now we equate the coefficients.
03:55
Right.
03:56
So we're going to say, equating coefficients for x.
04:18
So we start with the simplest, right? i usually like to start from the back, right? so i know my constant is just minus 2c.
04:30
Right, minus 2c is equal to minus 20.
04:35
Right? so this means that c is equal to 10.
04:39
Hence, for the x term, i have c minus 2b is equal to six, right? so 10 minus 2b is equal to 6.
04:53
So minus 2b is equal to minus 4.
04:58
So b is equal to 2.
05:01
Right.
05:02
And lastly, for the a term, instead of using the x squared term, i can use the x cubed term because this is simpler, right? so i'm going to say a x cubed is equal to x cubed, right? so a is equal to 1.
05:22
And you can actually use the x squared term to prove that everything is correct...